Your BBK would be a bit large for any case based on the belted H&H...that's personal opinion, of course. Because of the magazine length...3.85" I think...you could have a long throated .375 H&H or a long throated 416 Remington, which would provide loading flexibility with long, heavy bullets and still let you shoot factory ammo if need be. Your action would not require any mods to accommodate these cartridges.

Otto